Optimizing an NBA Team’s Approach to Free Agency using a Multiple-Choice Knapsack Model

Abstract: Optimizing an NBA team’s approach to signing free agents can be viewed as a knapsack problem for which the available free agents comprise the items that can be placed into the knapsack, and each team’s cap space corresponds to the size of the knapsack. The salary cap presents teams with the problem of how to maximize the value of a fixed expenditure in a dynamic market. As free agents are signed, the talent pool available to fill a team’s needs decreases. However, as the amount of available money in the market decreases, the amount required to sign a given player also decreases. Thus, teams are faced with the dilemma of when to offer a contract and how much money to offer. The optimal strategy for solving this problem must consider the trade-off between losing a player to a competitor and acquiring the player at a price that preserves as much cap space as possible for additional players. This trade-off can be solved by a dynamic program based on the multiple-choice knapsack problem, which optimizes the benefit of signing a player at any particular point in the free agency period in comparison to the opportunity costs of the cap space required to do so.